Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 18, 2017; final manuscript received October 25, 2017; published online January 25, 2018. Assoc. Editor: Alper Erturk.

This paper proposes an isolation transmissibility for the bending vibration of elastic beams. At both ends, the elastic beam is considered with vertical spring support and free to rotate. The geometric nonlinearity is considered. In order to implement the Galerkin method, the natural modes and frequencies of the bending vibration of the beam are analyzed. In addition, for the first time, the elastic continuum supported by boundary springs is solved by direct numerical method, such as the finite difference method (FDM). Moreover, the detailed procedure of FDM processing boundary conditions and initial conditions is presented. Two numerical approaches are compared to illustrate the correctness of the results. By demonstrating the significant impact, the necessity of elastic support at the boundaries to the vibration isolation of elastic continua is explained. Compared with the vibration transmission with one-term Galerkin truncation, it is proved that it is necessary to consider the high-order bending vibration modes when studying the force transmission of the elastic continua. Furthermore, the numerical examples illustrate that the influences of the system parameters on the bending vibration isolation. This study opens up the research on the vibration isolation of elastic continua, which is of profound significance to the analysis and design of vibration isolation for a wide range of practical engineering applications.

The natural frequencies of the linear beam versus the stiffness of vertical support springs: (a) the first-order frequency, (b) the second-order frequency, (c) the third-order frequency, and (d) the fourth-order frequency

The stable steady-state amplitude of three different locations along the elastic beam versus the excitation frequency: the Galerkin truncate calculations: (a) the first-order resonance and (b) the third-order resonance

Comparisons of linear and nonlinear responses around the first natural frequency of the beam for various boundary spring stiffness: (a) the stable steady-state response and (b) the vibration transmissibility

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